Empirical descriptions and studies suggest that generally depositors observe a sample of previous decisions before deciding if to keep their funds deposited or to withdraw them. These observed decisions may exhibit different degrees of correlation across depositors. In our model depositors are assumed to follow the law of small numbers in the sense that they believe that a bank run is underway if the number of observed withdrawals in their sample is high. Theoretically, with highly correlated samples and infinite depositors runs occur with certainty, while with random samples it needs not be the case, as for many parameter settings the likelihood of bank runs is less than one. To investigate the intermediate cases, we use simulations and find that decreasing the correlation reduces the likelihood of bank runs, often in a non-linear way. We also study the effect of the sample size and show that increasing it makes bank runs less likely. Our results have relevant policy implications.